ICAMM 2014, Nantes, France, 08/07/2014

Non-adiabatic effects in the 1D optical spectra of H

2

+ and H2

Alison CrawfordNano-Bio Spectroscopy Group and ETSF Scientific Development Center,

Departamento de Física de Materiales,Centro de Física de Materiales CSIC-UPV/EHU-MPC and DIPC,

Universidad del País Vasco UPV/EHU, Avenida de Tolosa 72, E-20018, San Sebastián, Spain

http://nano-bio.ehu.es/users/alison

Outline● Motivations

● Model systems: H2

+ charged and H2 neutral in 1D

● Configurations in 1D

● Optical spectra: theory

● Results: classical fixed/moving vs quantum ions

● Conclusions

Motivations● Optical absorption spectra

● Quantum (feasible in 1D for one- and two-electron systems)

● See effects of quantum ions vs classical fixed/moving ions

● Adiabatic approximations → decouple electron-ion

– Born Oppenheimer Approximation (BOA)

● Electron: ● Ion:

not in 1D

Classical negligible if >

Model systems: H2

+ and H2

in 1D

● H2

+ (centre of mass) → simplify to 2 variable problem in 1D

● H2 (centre of mass) → simplify to 3 variable problem in 1D

Change only mass→ → assess electron-ion coupling

M me

X=X 2−X 1

ξ=x−X1+X 2

2

H internal(X , x ,ξ)=−1M

∂2

∂X 2−

1+M4M

∂2

∂ξ2− ∂2

∂ x2−

1

√( X2 +x2+ξ)

2

+ΔIe2

−1

√( X2 +x2−ξ)

2

+ΔIe2

−1

√( X2 −x2+ξ)

2

+ΔIe2

−1

√( X2 −x2−ξ)

2

+ΔIe2

+1

√x2+Δee2+

1

√X 2+ΔII2

x=x2−x1

ξ=x1+x2

2−X 1+X 2

2

M H 2+ ;M D2

+ ;M Li2+ ;M K2

+

H internal(X ,ξ)=−1M

∂2

∂ X 2−

2M+14M

∂2

∂ξ2−

1

√( X2 +ξ)2

+ΔIe2

−1

√( X2 −ξ)2

+ΔIe ²

+1

√X 2+ΔII2

Configurations in 1D→ 2D and 3D bare Coulomb→ Particles confined in 1D trajectories separated by → Classical geometrical interpretation

Δ

Optical spectra → theory● Kicked initial state in the dipole approximation

● Dipole moment

● Optical spectra

d (t)=⟨ψ∣ξ∣ψ ⟩

σ(ω)=limK→∞

4 παω

Kℑ[∫

0

∞

dt e−iω t (d (t )−d (0))]

ψ(X ,ξ ,t )≈ψgse−iεgs t

2 +iK∑k>0

e−iεk t

2 ⟨ψk∣(2M+22M+1

)ξ∣ψgs⟩ψk

V ext=eiK ξ≈(1+iK ξ)

ψ(X , x ,ξ , t )≈ψgse−iεgs t

2 +iK∑k>0

e−iεk t

2 ⟨ψk∣−2(2M+1)+2M(M+1)(2M+1)(M+1)

ξ∣ψgs⟩ψkODD !! EVEN

ODD

Optical spectra H2

+: classic vs quantum

Overall shape and Xeq

G. Sansone et al. Nature, Vol. 465, 2010 E0(X

eq) → E

i(X

eq) electronic

Even to odd splitting Asymmetry (repulsion)

Fix X (no t) Quantum vs fixed Xeq

(with t)

Optical spectra H2

+: classic move/fix

Ionic motion peak

Vanishes MFixed/moving not capture

quantum non-adiabatic effects

↑

d(t)

t

Optical spectra H2

+: classic vs quantum

Even to Odd splits Positronium first peak

Larger M third/fifth peak Width/Lifetime → ele to ion M (not elastic)↓

d(t)

t

Optical spectra H2: classic vs quantum

Overall PES shape and equilibrium geometry

ee repulsion dissociate 2H

G. Sansone et al. Nature, Vol. 465, 2010 Weaker effects

Repulsion (asymmetry)

Optical spectra H2: classic vs quantum

Symmetry x (even) (odd)

Gs → 2Ex Gs → 6Ex

Gs → 10 Ex

x

ξ

Planar alsoGs → 3Ex

both odd ??

ξ

Conclusions● Non adiabatic electron-ion effects

– Splitting + asymmetry (non-elastic)

– Stronger for charged H2

+

– Configuration → repulsive interionic potential

– Vanish as mass increases (relative strengths)

● Range of applicability of ED and BOA

● Compare 1D vs 3D optical spectra (only PES)

● Analytical interpretation of the non-adiabatic features

ICAMM 2014, Nantes, France, 08/07/2014

Non-adiabatic effects in the 1D optical spectra of H

2

+ and H2

Alison CrawfordNano-Bio Spectroscopy Group and ETSF Scientific Development Center,

Departamento de Física de Materiales,Centro de Física de Materiales CSIC-UPV/EHU-MPC and DIPC,

Universidad del País Vasco UPV/EHU, Avenida de Tolosa 72, E-20018, San Sebastián, Spain

http://nano-bio.ehu.es/users/alison

THANK YOU

1st approach ● BOA analysis (ion dark coupling)

?

i ℏ ∂∂ t

χi(R , t )=(−ℏ

2

2μ I∂

2

∂R2−F i(R−Ri)+εi(Ri))χi(R ,t )

E0 χ0(R)=(−ℏ

2

2μ I∂

2

∂R2−12K (R−R0)

2+ε0(R0))χ0(R)

d (t )∝∫ dRci(t )χ0*(R)χ i(R ,t)

2nd approach ● Two-level system (ion bonding/antibonding)

belowabove

Eele

Eion E

ion

Eele